Spinors and the reference point of quasilocal energy

TL;DR

This paper explores how spinorial methods, specifically the Witten-Nester integral, relate to and determine the reference point of Brown-York quasilocal energy, especially in Schwarzschild spacetime.

Contribution

It demonstrates that Sen-Witten spinors can naturally set the reference point for Brown-York energy, linking spinorial expressions to quasilocal energy definitions.

Findings

Witten-Nester integral evaluated on solution spinors matches Brown-York energy.

Sen-Witten spinors determine the flat-space reference point in Schwarzschild geometry.

Proposes a similar approach for Dougan-Mason quasilocal energy.

Abstract

This paper investigates the relationship between the quasilocal energy of Brown and York and certain spinorial expressions for gravitational energy constructed from the Witten-Nester integral. A key feature of the Brown-York method for defining quasilocal energy is that it allows for the freedom to assign the reference point of the energy. When possible, it is perhaps most natural to reference the energy against flat space, i.e. assign flat-space the zero value of energy. It is demonstrated that the Witten-Nester integral when evaluated on solution spinors to the Sen-Witten equation (obeying appropriate boundary conditions) is essentially the Brown-York quasilocal energy with a reference point determined by the Sen-Witten spinors. For the case of round spheres in the Schwarzschild geometry, these spinors determine the flat-space reference point. A similar viewpoint is proposed for the Schwarzschild-case quasilocal energy of Dougan and Mason.